It seems the main assumption of "I think therefore I am" is the "I". By the time anyone is philosophising in such a degree of sophistication the questioning entity is already enmeshed and integrated in a cultural network: Being & Time ¶ 25. An Approach to the Existential Question of the "Who" of Dasein:-
In clarifying Being-in-the-world we
have shown that a bare subject without a world never 'is' proximally,
nor is it ever given. And so in the end an isolated "I" without Others
is just as far from being proximally given. If, however, 'the Others'
already are there with us [mit da sind] in Being-in-the-world, and
if this is ascertained phenomenally, even this should not mislead us
into supposing that the ontological structure of what is thus
'given' is obvious, requiring no investigation.
Even though local interests are ascribed to the "I" the being of the "I" is a networked entity formed in a society, so by the time thought questions existence it is socialised thought.
Essentially, the doubting Descartes of "dubito, ergo cogito, ergo sum" cannot even be certain of his body. In doubting, there is thought, therefore there is existence of thought.
In the Heideggerian œuvre the "I" is a part of the being of Dasein that involves ownership (Jemeinigkeit) but it is not necessarily all of Dasein. However, the being of Dasein cannot be fully explained in thought and language because it is what produces thought in the first place. It is in the objective conception of the "I" formed retrospectively that the questionable assumption of the OP's question can arise.
The way Heidegger relates the "I" to Dasein (the primary mode of which is Being-in-the-world) is indicated in Being & Time ¶ 26. The Dasein-with of Others and Everyday Being-with:-
Thus in characterizing the encountering of Others, one is again
still oriented by that Dasein which is in each case one's own. But
even in this characterization does one not start by marking out and
isolating the 'I' so that one must then seek some way of getting over
to the Others from this isolated subject? To avoid this
misunderstanding we must notice in what sense we are talking about
'the Others'. By 'Others' we do not mean everyone else but me—those
over against whom the "I" stands out. They are rather those from whom,
for the most part, one does not distinguish oneself—those among whom
one is too. This Being-there-too [Auch-da-sein] with them does not
have the ontological character of a
Being-present-at-hand-along-'with' them within a world. This 'with'
is something of the character of Dasein; the 'too' means a sameness of
Being as circumspectively concernful Being-in-the-world. 'With' and
'too' are to be understood existentially, not categorially. By
reason of this with-like [mithaften] Being-in-the-world, the world
is always the one that I share with Others. The world of Dasein is a
with-world [Mitwelt].
So when Descartes deduces his existence (as "ergo sum") that existential Dasein already extends to the with-world in which he learned to think (with others). He is not isolated in a solipsistic way to the "I"; he discovers a cultivated intelligence.